Singular conformally invariant trilinear forms and generalized Rankin Cohen operators
aa r X i v : . [ m a t h . R T ] A p r Singular conformally invariant trilinear forms andgeneralized Rankin Cohen operators
Ralf Beckmann Jean-Louis ClercApril 18, 2011
Dedicated to Elias Steinon the occasion of his 80th birthday
Abstract
The most singular residues of the standard meromorphic family oftrilinear conformally invariant forms on C ∞ c ( R d ) are computed. Theirexpression involves covariant bidifferential operators (generalized RankinCohen operators), for which new formulæ are obtained. The main toolis a Bernstein-Sato identity for the kernel of the forms. Introduction
Let E be a finite dimensional Euclidean space of dimension d, ( d ≥ . Let G ≃ SO (1 , d + 1) be the (connected component of) the conformal group of E , acting by rational transformations on E . Much interest has been devotedto various invariant or covariant objects for this action. There is a naturalaction of G on a space of densities on E . Identifying the densities withfunctions on E , the action is given by π λ ( g ) f ( x ) = κ ( g − , x ) ρ + λ f ( g − ( x )) , where κ ( g, x ) is the conformal factor (the infinitesimal rate of dilation) ofthe transformation g ∈ G at x ∈ E , λ is a complex parameter and ρ = d .Let λ , λ , λ be three complex numbers. A continuous trilinear form L on The case d = 1 could be treated along the same lines, but there are some differences,which would require separate statements. See [15] for a study of this case. ∞ c ( E ) × C ∞ c ( E ) × C ∞ c ( E ) is said to be conformally invariant with respectto π λ ⊗ π λ ⊗ π λ , if, for any three functions f , f , f ∈ C ∞ c ( E ) L ( π λ ( g ) f , π λ ( g ) f , π λ ( g ) f ) = L ( f , f , f )where g is in G (strictly speaking, defined on the union of the supportsof the three functions). These trilinear forms have been investigated in aprevious work of the second author in collaboration with B. Ørsted (see [3]).Generically, for λ = ( λ , λ , λ ) in C , there is a unique (up to a multiple)such invariant trilinear form. Viewing the trilinear form as a distributionon E × E × E , it has a smooth density on the open set { ( x , x , x ) ∈ E × E × E ; x = x , x = x , x = x } , given by l β ( x , x , x ) = | x − x | β | x − x | β | x − x | β where β = ( β , β , β ) is a triplet of complex numbers, uniquely determinedby λ = ( λ , λ , λ ) (see (23)). The corresponding distribution L β is definedby meromorphic continuation, and has simple poles along certain planes in C . The study of the residues was begun by the second author in [2], and thepresent paper deals with the most singular residues. They are distributionssupported on the diagonal D = { ( x, x, x ) ; x ∈ E } . They turn out to beintimately related with covariant bidifferential operators , that is differentialoperators from C ∞ c ( E × E ) into C ∞ c ( E ) which satisfy a relation of the form D ( π λ ( g ) f ⊗ π µ ( g ) f ) = π ν ( g ) D ( f ⊗ f )for some ( λ, µ, ν ) ∈ C and for any g ∈ G, f , f ∈ C ∞ c ( E ). Such oper-ators have a long history (see [8], [9]), and the most celebrated ones arethe Rankin Cohen operators , which are holomorphic bidifferential operatorson the complex upper half plane and covariant under the group
P SL ( R ).In the context of conformal geometry, they were studied by Ovsienko andRedou ([17]). See also [13], [14].The basic ingredient we use for computing the residues is a Bernstein-Sato identity (cf Theorem 4.1). Recall that for f , f , f three arbitrarynonnegative polynomials on E , there exists a differential operator B = B ( x , x , x , s , s , s , ∂ x , ∂ x , ∂ x ) on E which is polynomial in the x j and s j , and a polynomial b on C , such that B ( f s +11 f s f s ) = b ( s , s , s ) f s f s f s . (1)2uch identities exist in general (cf [19], [20]), but their explicit determina-tion is seldom known. Once such an identity is known, the computationof the residues is easy. As a consequence, we find new expressions for thecovariant bidifferential operators alluded to previously. More general resultson Bernstein-Sato identities will appear in [1].The plan of the paper is as follows. Section 1 collects some results ondistributions supported on a subspace. Section 2 is an elementary approachto computing the residues, much in the spirit of Gelfand and Shilov ([10]).In particular it allows to determine the residues along the ”first” plane ofpoles. Section 3 exploits the invariance property of the trilinear forms andthe covariance property of the associated bidifferential operators, makingconnection with the results of [17]. Section 4 is devoted to the Bernstein-Sato identity (Theorem 4.1), which is used in section 5 to give a formulafor the residues, and as a consequence, a new expression for the covariantbidifferential operators (see (35)). Section 6 presents several remarks andperspectives on the subject.The second author wishes to dedicate the present paper to Elias Stein,on the occasion of his 80th birthday. Let E be a finite dimensional real vector space, and let V be a linear subspaceof E . Let E ′ be the dual space of E , and let V ⊥ = { ξ ∈ E ′ ; ξ | V = 0 } . Let u be a distribution on V . The assignment C ∞ c ( E ) ∋ ϕ ( u, ϕ | V )defines a distribution on E , the natural extension of u , hereafter denoted by e u . Clearly Supp ( e u ) = Supp ( u ) ⊂ V . We now characterize the wavefront set of e u (cf [11]). Proposition 1.1.
Let u be in D ′ ( V ) , and let e u the associated distributionon E . Then W F ( e u ) = { ( x, ξ ) ∈ Supp ( u ) × ( E ′ \
0) ; ξ ∈ V ⊥ or ( x, ξ | V ) ∈ W F ( u ) } . (2)3 roof. Choose a subspace W such that E = V ⊕ W . For ξ ∈ E ′ , let ξ = ξ ′ + ξ ′′ , where ξ ′ ∈ W ⊥ and ξ ′′ ∈ V ⊥ . Let ϕ be in C ∞ c ( E ). Then ϕ e u isa distribution with compact support and its Fourier transform is given by F ( ϕ e u )( ξ ) = ( u, e − i ( ξ ′ ,. ) ϕ | V ) . Let ( x , ξ ) be in the set described by the RHS of (2). If ξ ′ = 0, thenchoose ϕ such that h u, ϕ | V i 6 = 0 (which is always possible since x belongsto Supp ( u )), so that F ( ϕ e u )( ξ ) cannot decrease rapidly in a conic neigh-bourhoud of ξ . If ( x , ξ ′ ) belongs to W F ( u ), F ( ϕ e u )( ξ ) cannot decreaserapidly on a conic neigbourhood of ξ ′ in W ⊥ \ a fortiori on a conic neig-bourhood of ξ in E ′ \
0. Conversely, assume ( x , ξ ′ ) does not belong to W F ( u ) and ξ ′ = 0. Then, for ϕ with a sufficently small support near x and for ξ in a (small enough) conic neigbourhood of ξ , F ( ϕ e u )( ξ ) can bedominated by C N (1 + | ξ ′ | ) − N for any integer N . But in a (sufficently small)conic neighbourhood of ξ ′ one has | ξ | ≤ C | ξ ′ | for some constant C >
0, sothat F ( ϕ e u )( ξ ) is dominated by C N (1+ | ξ | ) − N . Hence ( x , ξ ) / ∈ W F ( e u ).To further investigate distributions supported on V , one needs to intro-duce normal derivatives . Fix a splitting E = V ⊕ W as above, and chooseccordinates w , w , . . . , w p on W , which can be regarded as (a partial set of)coordinates on E by extending them by 0 on V . Let I = ( i , . . . , i p ) be ap-tuple of natural integers, let | I | = i + i + · · · + i p . Let D I be the operator(the D I ’s are often referred to as normal derivatives ), defined by D I ϕ ( v ) = ∂ | I | ϕ∂w i . . . ∂w i p p ( v ) , mapping smooth functions on E to smooth functions on V . To any distri-bution u on V , one can associate the distribution D I e u defined by( − | I | ( D I e u, ϕ ) = ( u, D I ϕ ) . Observe that
W F ( D I e u ) = W F ( e u ). The inclusion ⊂ is obvious, whereas theopposite inclusion is obtained by testing against functions ϕ of the form ϕ ( v, w ) = χ ( v ) w I ψ ( w ) , (3)where χ ∈ C ∞ c ( V ), w I = w i . . . w i d d and ψ is a function in C ∞ c ( W ) which isidentically equal to 1 in a neigbourhood of 0.4ow let U be a distribution on E , with Supp ( U ) ⊂ V . The structuretheorem of L. Schwartz asserts that there exist distributions u I on V suchthat U = X I D I e u I , where the sum is locally finite. Moreover, the u I ’s are unique.If all the distributions u I are given by smooth densities, then from (2), W F ( U ) ⊂ E × ( V ⊥ \ Proposition 1.2.
Let U be a distribution supported in V , and assume that W F ( U ) ⊂ V × ( V ⊥ \ . Then there exist smooth functions u I on V such that, for any ϕ ∈ C ∞ c ( E )( U, ϕ ) = Z V X I u I ( v ) D I ϕ ( v ) dv . Proof.
By the previous result, U = P I D I e u I , where u I is some distributionon V . The assumption on the wavefront set of U , when tested againstthe functions of the form given by (3) implies that, for each d -tuple I , W F ( e u I ) ⊂ V × ( V ⊥ \ W F ( u I ) = ∅ by Proposition 1.1.As the projection onto the first coordinate of the wavefront set is preciselythe singular support, each u I coincides with a smooth function on E .A transverse differential operator D is a mapping form C ∞ c ( E ) in C ∞ c ( V )which is given by Dϕ ( v ) = X I a I ( v ) D I ϕ ( v ) , where D I are the normal derivatives introduced earlier, and the a I ’s aresmooth functions on V . The sum is always assumed to be locally finite. No-tice that the a I are well determined, again by testing the operator againstfunctions of the form given by (3). The previous proposition can be refor-mulated as : any distribution U supported on a linear subspace V , such that W F ( U ) ⊂ V × ( V ⊥ \
0) can be realized as(
U, ϕ ) = Z V Dϕ ( v ) dv , (4)for some transverse differential operator D . Moreover, (once a splitting of E as V ⊕ W has been chosen) D is uniquely determined .Invariance properties of a singular distribution are reflected in the asso-ciated transverse differential operator. Here is a special case, fitted for ourneeds. 5 roposition 1.3. Let U be in D ′ ( E ) , supported on V . Assume that U isinvariant under translations by elements of V . Then (cid:0) U, ϕ (cid:1) = Z V Dϕ ( v ) dv , where D is a transverse differential operator with constant coefficients .Proof. Let v be any element of V \
0, and let X v be the vector field on E which is constant and equal to v at each point of E . Then, the invarianceproperty of U amounts to the equalities X v U = 0 for any v ∈ V . Hence, by[11] Theorem 8.3.1 W F ( U ) ⊂ { ( x, ξ ) ∈ V × ( E ′ \ , ξ ( v ) = 0 } . As this is valid for any v ∈ V , W F ( U ) ⊂ { ( x, ξ ) ∈ V × ( V ⊥ \ } . By Proposition 1.2, U is given by a transverse differential operator D , i.e.( U, ϕ ) = Z V Dϕ ( v ) dv where Dϕ ( v ) = P I a I ( v ) D I ϕ ( v ). Now, for any v ∈ V , X v commutes withany D I , such that, by integration by parts,0 = ( U, X v ϕ ) = − Z V X I X v a I ( v ) D I ϕ ( v ) dv . Now fix a d -tuple I , and check this equality on functions of the form (3). Ityields X v a I = 0 for any v ∈ V and hence a I is a constant. Remark . All the results of this section could be formulated for distributionssupported on a closed submanifold.
In this section, we consider the standard Euclidean space E = R d and denotethe distance of two points x, y ∈ E by | x − y | . Let β be a complex numberand let l β ( x, y ) = | x − y | β . ℜ β large enough, the kernel l β is locally integrable on E × E , so that itdefines a distribution on E × E . It can be extended meromorphically (as adistribution), having simple poles at the points β = − d − k, k ∈ N (see e.g.[10]).Now let β = ( β , β , β ) be in C . Set, for x , x , x ∈ El β ( x , x , x ) = l β ( x , x ) l β ( x , x ) l β ( x , x ) . For f in C ∞ c ( E × E × E ) the integral formula L β ( f ) = Z E × E × E f ( x , x , x ) l β ( x , x , x ) dx dx dx is well defined for ℜ ( β j )( j = 1 , ,
3) large enough and can be meromorphi-cally continued in C as a distribution on E × E × E . Theorem 2.1.
The map β β can be meromorphically extended to C , with simple poles along the four families of planes defined by one of thefollowing equations β = − d − k , β = − d − k , β = − d − k (5) β + β + β = − d − k (6) where k , k , k , k ∈ N . The analoguous result for the sphere was obtained in a joint work of thesecond author with B. Ørsted [3], and the proof for the flat case requiresonly minor modifications. In fact the flat case and the spherical case arerelated by a stereographic projection already used in [3].A pole is said to be of first type if it satisfies one of the equations (5), of second type if it satisfies one of the equations (6).For poles of the first type, the residues were studied in [2]. We concen-trate on poles of the second type. For each k ∈ N , let H k be the planedefined by H k = { β ∈ C ; β + β + β = − d − k } . In [3] was already proven that, at a generic point (see precise statementbelow) of such a plane, the residue (viewed as a distribution on E × E × E )is supported on the diagonal D = { ( x, x, x ) | x ∈ E } . This will be reproved, in a more elementary and explicit way (see Theorem2.4). 7he distribution L β is invariant by diagonal translations, i.e. by map-pings t v , v ∈ E , where t v ( x , x , x ) = ( x + v, x + v, x + v ) . Clearly the residue at some pole β will have the same invariance. So, for β a pole of the second type, we are in the situation of Proposition 1.3. Onehas to choose a normal space to D in E × E × E , and our choice will be W = { (0 , y, z ) , y ∈ E, z ∈ E } . The concept of transverse differential operator introduced in the previoussection can be reinterpreted in this context. A bidifferential operator D is amap from smooth functions on E × E to smooth functions on E of the form Dϕ ( v ) = X I,J a IJ ( v ) ∂ | I | + | J | ϕ∂y I ∂z J ( v, v ) , where I and J are d -tuples of integers, y j (resp z j ) are coordinates on thefirst factor (resp. second factor), associated to a choice of a basis of E andthe a IJ ’s are smooth functions on E . The sum is assumed to be locallyfinite. When the a IJ are constant functions (hence all 0 except for a finitenumber), the operator is said to be with constant coefficients. Theorem 2.2.
Let β ∈ H k for some k ∈ N , but such that none of theequations (5) is satisfied. Then there exists a unique bidifferential operatorwith constant coefficients D β such that, for f in C ∞ c ( E ) and g ∈ C ∞ c ( E × E ) Res ( L β , β )( f ⊗ g ) = Z E f ( x ) D β g ( x ) dx . (7)Here f ⊗ g stands for the function E × E × E ∋ ( x, y, z ) f ( x ) g ( x, y ) . Needless to say, as those functions are dense in C ∞ c ( E × E × E ), (7) determinescompletely the residue.As already observed, the distribution L β is invariant by any diagonaltranslation. To take adavantage of this, define for ϕ ∈ C ∞ c ( E × E × E )Φ( y, z ) = Z E ϕ ( v, y + v, z + v ) dv . (8) As symmetry among the three variables is broken, from now on, we use ( x, y, z ) fornotation of an element in E × E × E . ϕ in C ∞ c ( E × E × E ), the integral converges and defines a function Φwhich belongs to C ∞ c ( E × E ). Moreover, the correspondance ϕ Φ iscontinuous. Notice for further reference thatΦ(0 ,
0) = Z E ϕ ( x, x, x ) dx . (9) Lemma 2.1.
Assume that ℜ ( β j ) > − d, j = 1 , , and ℜ ( β + β + β ) > − d . Then, for any ϕ ∈ C ∞ c ( E × E × E ) L β ( ϕ ) = Z E × E | y | β | z | β | y − z | β Φ( y, z ) dy dz . Proof.
The conditions on the parameter β guarantee the convergence of theintegrals. The equality of the integrals is obtained through the affine changeof variables v = x , y = x − x , z = x − x . Let Σ = { ( σ, τ ) ∈ E × E, | σ | + | τ | = 1 } be the unit sphere in E × E , and denote by dµ the Lebesgue measure on Σ.Recall the integration formula in polar coordinates Z E × E Φ( y, z ) dy dz = Z ∞ Z Σ Φ( rσ, rτ ) dµ ( σ, τ ) r d − dr (10) Lemma 2.2.
For ψ in C ∞ c (Σ) let I β ( ψ ) = Z Σ | σ | β | τ | β | σ − τ | β ψ ( σ, τ ) dµ ( σ, τ ) . (11) i ) Assume that ℜ ( β j ) > − d for j = 1 , , . Then the integral (11) isconvergent and defines a distribution I β . ii ) The map β β can be extended meromorphically to C , withsimple poles along the family of planes given by the following equations : β j = − d − k j , j = 1 , , , k j ∈ N . Proof.
The three subsets of Σ { ( σ, τ ) ∈ Σ , σ = 0 } , { ( σ, τ ) ∈ Σ , τ = 0 } , { ( σ, τ ) ∈ Σ , σ = τ } disjoint submanifolds of dimension d − d )in Σ. Recalling that we assumed ℜ ( β j ) > − d , for j = 1 , ,
3, the integrals Z Σ | σ | β dµ ( σ, τ ) , Z Σ | τ | β dµ ( σ, τ ) , Z Σ | σ − τ | β dµ ( σ, τ )are convergent and hence the integral I β is convergent by applying a suit-able argument involving a partition of unity. This shows i ). Similarly, themeromorphic extension and the location of poles (also the fact that the polesare simple) are classical and can be easily deduced from [10].Let Φ be a function in C ∞ c ( E × E ). For r in R , let ψ r be the function onΣ defined by ψ r ( σ, τ ) = Φ( rσ, rτ ) , ( σ, τ ) ∈ Σ . (12)Then ψ r belongs to C ∞ (Σ) and the map ( r, Φ) ψ r is continuous from R × C ∞ c ( E × E ) to C ∞ (Σ). Lemma 2.3.
Assume that ℜ ( β j ) > − d, j = 1 , , and ℜ ( β + β + β ) > − d . Then, for any ϕ ∈ C ∞ c ( E × E × E ) L β ( ϕ ) = Z ∞ r d − β + β + β I β ψ r dr . (13)This is just using the formula for integration in polar coordinates. Lemma 2.4.
Let γ be in C ∞ c ( R ) and assume that γ is an even function.Then the integral I s ( γ ) = Z ∞ r s γ ( r ) dr = 12 Z + ∞−∞ | r | s γ ( r ) dr is convergent for ℜ s > − . The map s I s ( γ ) can be extended mero-morphically to C with simple poles at s = − − k, k ∈ N . Moreover, theresidues at the poles are given by Res ( I s ( γ ) , − − k ) = 1Γ(2 k + 1) (cid:0) ddr (cid:1) k γ (0) . (14)For a proof, see [10]. Theorem 2.3.
Let k ∈ N and let β = ( β , β , β ) satisfy the followingassumptions : i ) β + β + β = − d − k i ) β j / ∈ − d − N .Let ϕ be a function in C ∞ c ( E × E × E ) and form successively the functions Φ defined by (8) and ψ r defined by (12) . The function β β ( ϕ ) has aresidue at β given by Res ( L β ( ϕ ) , β ) = 1Γ(2 k + 1) (cid:0) ddr (cid:1) k | r =0 I β ( ψ r ) . (15) Proof.
Observe that I β ψ r is well defined (lemma 2.2) and, as a functionof r is easily seen to be in C ∞ c ( R ). Moreover, the distribution I β is even,whereas ψ − r ( σ, τ ) = ψ r ( − σ, − τ ), hence I β ψ r is an even function of r . Nowlet γ = I β ψ r . Then (13) can be rewritten as L β ( ϕ ) = I s ( γ ). Observe that2 d − β + β + β = − − k and eventually apply (14) to conclude.The expression obtained for the residue (viewed as a distribution on E × E × E ) shows that it is supported by the diagonal D . In fact, if ϕ ∈ C ∞ ( S )vanishes on a neighbourhood of D , then Φ vanishes in a neigbourhood of(0 ,
0) in E × E , and hence ψ r vanishes identically for | r | small enough,so that I β ( ψ r ) = 0 for | r | small enough. Hence the residue (evaluatedagainst ϕ ) at β vanishes. Now, the structure of distributions supported bya submanifold is known from Schwartz’s theorem. This requires choosingat each point ( x, x, x ) of D a complementary subspace to the tangent space(”normal coordinates”) at D . The choice will be N x,x,x = { ( x, y, z ) , y ∈ E, z ∈ E } . Moreover, as we are interested in the trilinear form rather than the distri-bution, the space of test functions will be restricted to functions of the form( f ⊗ g ) ( x, y, z ) = f ( x ) g ( y, z ) , x ∈ E, ( y, z ) ∈ E × E .With this change of point of view and notation, let us write more explic-itly (15). We will use the following convention for coordinates on E × E :for 1 ≤ i ≤ d , let v i = y i , if 1 ≤ i ≤ d, v i = z i − d , if d + 1 ≤ i ≤ d . and similarly on Σ ρ i = σ i , if 1 ≤ i ≤ d ρ i = τ i − d , if d + 1 ≤ i ≤ d . Theorem 2.4.
Let β satisfy the same assumptions as in Theorem 2.3. Let f ∈ C ∞ c ( E ) and g ∈ C ∞ c ( E × E ) . Then Res ( L β ( f ⊗ g ) , β ) = 1Γ(2 k + 1) Z E f ( x )( D β g )( x ) dx , (16)11 here D β is the bidifferential operator with constant coefficients given by D β g ( v ) = X ≤ i ,i ,...,i k ≤ d a i ,i ,...,i k ( β ) ∂ k g∂v i ∂v i . . . ∂v i k ( v, v ) where a i ,i ,...,i k ( β ) = Z Σ ρ i . . . ρ i k | σ | β | τ | β | σ − τ | β dµ ( σ, τ ) . (17) Proof.
First, for ( σ, τ ) in Σ, ψ r ( σ, τ ) = Z E f ( v ) g ( v + rσ, v + rτ ) dv so that (cid:0) ddr (cid:1) k | r =0 ψ r ( σ, τ ) = Z E f ( v ) R (2 k ) σ,τ g ( v ) dv , where R (2 k ) σ,τ is the bidifferential operator given by R (2 k ) σ,τ g ( v ) = X ≤ i ,i ,...,i k ≤ d ρ i ρ i . . . ρ i k ∂ k g∂v i ∂v i . . . ∂v i k ( v, v ) . Hence
Res ( L β ( f ⊗ g ) , β ) = (cid:0) ddr (cid:1) k | r =0 I β ( ψ r ) = I β (cid:16)(cid:0) ddr (cid:1) k | r =0 ψ r (cid:17) = I β (cid:0) Z E f ( v ) R (2 k ) σ,τ g ( v ) dv (cid:1) = Z E f ( v ) D β g ( v ) dv where D β is the bidifferential operator given by D β g ( v ) = X ≤ i ,i ,...,i k ≤ d a i ,i ,...,i k ( β ) ∂ k g∂v i ∂v i . . . ∂v i k ( v, v )where a i ,i ,...,i k ( β ) = Z Σ ρ i . . . ρ i k | σ | β | τ | β | σ − τ | β dµ ( σ, τ ) , with the same convention as above.Of course these integrals are to be understood in the sense of distribu-tions, obtained by meromorphic continuation.12hen k = 0, there is only one term, so D β = c β Id and it is possibleto evaluate the constant c β . Proposition 2.1.
Assume that β + β + β = − d , and assume that β j / ∈− d − N , j = 1 , or . Then Res ( L β ( f ) , β ) = c ( β ) Z E f ( x ) dx (18) where c ( β ) = π d (2 √ d Γ(2 d )Γ( d ) Γ( β + d )Γ( − β − d ) Γ( β + d )Γ( − β − d ) Γ( β + d )Γ( − β − d ) . (19) Proof.
Let c be the stereographic projection, defined from E into the sphere S of radius 1 in R d +1 by c ( x ) = −| x | | x | x | x | ... x d | x | . The map c is conformal, and more precisely, for any tangent vector ξ ∈ R d , | Dc ( x ) ξ | = 21 + | x | | ξ | which implies Z S f ( σ ) dσ = 2 d Z E f ( c ( x ))(1 + | x | ) − d dx for f any integrable function on S . Moreover, for any x, y in E , | c ( x ) − c ( y ) | = 2 | x − y | (1 + | x | ) / (1 + | y | ) / . By the same change of variables, the trilinear form L β is related to thetrilinear form K α on S studied in [3] through the relation K α (1 ⊗ ⊗
1) = 2 α + α + α +3 ρ L β ( f ) (20)where α j = β j + ρ for j = 1 , ,
3, and f is the function on E × E × E givenby f ( x , x , x ) = (1 + | x | ) − β β (1 + | x | ) − β β (1 + | x | ) − β β . L β ( f ) = (cid:0) √ π √ (cid:1) d Γ( β + β + β + 2 d )Γ( β + d )Γ( β + d )Γ( β + d )Γ( β + β + d )Γ( β + β + d )Γ( β + β + d ) (21)Hence, Res ( L β ( f ) , β ) = (cid:0) √ π √ (cid:1) d Γ( β + d )Γ( β + d )Γ( β + d )Γ( β + β + d )Γ( β + β + d )Γ( β + β + d ) . Now Z E f ( x, x, x ) dx = Z E (1 + | x | ) − ( β + β + β ) dx = Z E (1 + | x | ) − d dx = vol ( S d − ) Z ∞ (1 + r ) − d r d − dr = π d Γ( d )Γ(2 d )from which the lemma follows.In the general case (when k ≥ covariance properties of the bidifferential operator D β . Introduce the group of conformal transformations of E . A local transforma-tion Φ of E is said to be conformal if, at any point x where Φ is defined,and for any tangent vector ξ , | D Φ( x ) ξ | = κ ( x ) | ξ | . where κ is a smooth strictly positive function, called the conformal factorof Φ. Classically, to any element of the group G = SO (1 , d + 1), one canattach a rational conformal action on E . If d ≥
3, then this group exhauststhe group of positive local conformal diffeomorphisms (Liouville’s theorem).The group G operates globally on the sphere S = S d of dimension d (see[21]) and this action can be transferred to a (not everywhere defined) actionon E by using the stereographic projection. It can also be realized as the14roup generated by the translations, the rotations, the dilations and thesymmetry-inversion ι ι : x x | x | . To any element g ∈ G , let κ ( g, x ) be its conformal factor. Then κ satisfies acocycle relation, namely κ ( g g , x ) = κ ( g , g ( x )) κ ( g , x ) . A family of representations is associated to that cocycle. For λ in C , define π λ ( g ) f ( x ) = κ ( g − , x ) ρ + λ f ( g − ( x )) , where we set ρ = d . These representations are the noncompact realizationof the principal spherical series of SO (1 , d + 1) (cf [21]).Let λ = ( λ , λ , λ ) ∈ C . A continuous trilinear form L on C ∞ c ( E ) ×C ∞ c ( E ) ×C ∞ c ( E ) is said to be conformally invariant with respect to ( π λ , π λ , π λ )if, for three functions f , f , f in C ∞ c ( E ), for any g in a (sufficently small)neighbourhood of the neutral element in G , L (cid:0) π λ ( g ) f ⊗ π λ ( g ) f ⊗ π λ ( g ) f (cid:1) = L (cid:0) f ⊗ f ⊗ f (cid:1) . (22)Recall the main result of [3], which was stated for the action of thegroup G on the sphere, but the situations are essentially equivalent througha stereographic projection. Proposition 3.1.
Let β ∈ C and assume that none of the conditions (5) , (6) is satisfied. Let λ = ( λ , λ , λ ) be the unique element of C defined bythe equations β = − λ + λ + λ − ρβ = λ − λ + λ − ρβ = λ + λ − λ − ρ . (23) i ) The trilinear form L β is conformally invariant with respect to π λ ⊗ π λ ⊗ π λ . ii ) Any continuous trilinear form which is conformally invariant with respectto π λ ⊗ π λ ⊗ π λ is proportional to L β . By analytic continuation, the invariance is also valid for the residue of L β at some pole β . At poles of second type, this invariance property canin turn be translated in a covariance property for the bidifferential operator D β . 15 efinition 3.1. Let D be a bidifferential operator from C ∞ c ( E × E ) into C ∞ c ( E ) . Let λ, µ, ν be three complex numbers. Then D is said to be covariantwith respect to ( π λ ⊗ π µ , π ν ) if for any functions f ∈ C ∞ c ( E × E ) and g ina (small enough) neighbourhood of the neutral element in G , D ( π λ ( g ) ⊗ π µ ( g ) f ) = π ν ( g )( Df )Recall the following duality result for the representations π λ . Proposition 3.2.
Let λ ∈ C . Then, for any functions ϕ and ψ in C ∞ c ( E ) Z E π λ ( g ) ϕ ( x ) ψ ( x ) dx = Z E ϕ ( x ) π − λ ( g − ) ψ ( x ) dx . This duality result links together (singular) conformally invariant trilin-ear forms and covariant bidifferential operators.
Proposition 3.3.
Let D be a bidifferential operator from C ∞ c ( E × E ) into C ∞ c ( E ) . Let L be the continuous trilinear form defined for f ∈ C ∞ c ( E ) and g ∈ C ∞ c ( E × E ) by L ( f ⊗ g ) = Z E f ( x ) Dg ( x ) dx . Let λ, µ, ν be three complex numbers. Then the form L is invariant withrespect to ( π λ , π µ , π ν ) if and only if D is covariant with respect to ( π µ ⊗ π ν , π − λ ) . Corollary 3.1.
Let β = ( β , β , β ) ∈ H k for some k ∈ N , and such thatnone of the conditions (5) is satisfied. Let D β be the associated bidifferentialoperator given by Theorem 2.4. Let λ = ( λ , λ , λ ) be given by equations (23) . Then the bidifferential operator D β is covariant with respect to ( π λ ⊗ π λ , π λ + λ + ρ +2 k )This is a consequence of the expression of the residue Theorem 2.2,together with the duality result (Proposition 3.2). The fact that none ofthe conditions (5) is satisfied amounts to the conditions λ , λ / ∈ − k + N , λ + λ / ∈ − ρ − k − N . Covariant differential operators for the conformal group have been stud-ied intensively, and the following result was obtained sometimes ago by V.Ovsienko and P. Redou (see [17]). Recall the
Pochhammer’s symbol , for a acomplex number and m ∈ N ( a ) m = a ( a + 1)( a + 2) . . . ( a + m − . roposition 3.4. Let k be a nonnegative integer, and let λ, µ be complexnumbers. i ) Assume that λ, µ / ∈ { , − , − , . . . , − ( k − } . Then there exists a bidif-ferential operator D ( k ) λ,µ which is covariant with respect to ( π λ ⊗ π µ , π λ + µ + ρ +2 k ) . ii ) Assume moreover that λ, µ / ∈ {− ρ, − ρ − , . . . , − ρ − ( k − } . Thenthe operator is unique up to a constant. The operator D ( k ) λ,µ is explicitly described. The three fundamental bidif-ferential operators are ∆ y , ∆ z and the operator R defined for f in C ∞ ( E × E )by R ( f )( x ) = d X j =1 ∂ f∂y j ∂z j ( x, x ) . Then D ( k ) λ,µ = X r,s,t,r + s + t = k c rst ∆ ry R s ∆ tz where the c rst are explicitly determined coefficients, depending on λ, µ and k , namely c rst = ( − t − r r r ! (cid:18) r + s + tt (cid:19) ( s + 1) r ( λ + 1) rr X p =0 r ! t ! p ! ( λ + ρ + r − s + p ) t − p ( µ + ρ + s + 2 t ) r − p ( µ + 1) t − p (24)when r ≤ t , and for r ≥ t , c rst ( λ, µ ) = c tsr ( µ, λ ). Theorem 3.1.
Let β ∈ H k for some k in N . Assume moreover that noneof the conditions (5) are satisfied. Let λ = ( λ , λ , λ ) the unique solutionof the system (23) . Then ( λ , λ ) / ∈ { , − , − , · · · − ( k − } and D β = c β D ( k ) λ ,λ (25) for some constant c β .Proof. The conditions on β imply that λ , λ / ∈ − k + N , so that the con-ditions for the defintion of D ( k ) λ,µ are satisfied. Assume for a while that β is such that λ , λ / ∈ { , − ρ, − ρ − , . . . , − ρ − ( k − } . By the uniquenessstatement, there exists a constant c β such that D β = c β D ( k ) λ ,λ . By analyticcontinuation in the plane H k , this equality remains valid on the domainswere both sides are defined. 17or instance, if k = 1, D (1) λ,µ = − µ + ρλ + 1 ∆ y + 2 R − λ + ρµ + 1 ∆ z . (26)Theorem 3.1 however does not determinate the constant c β , and it seemsquite difficult to test the operator D ( k ) λ ,λ against the function f as we didfor the determination of the residue at a pole in H . A Bernstein-Sato identity (on the first parameter) is an identity of the form
B ℓ β +2 = b ( β ) ℓ β , where β + 2 = ( β + 2 , β , β ), B = B (( x, y, z ) , ∂ x , ∂ y , ∂ z , β ) is a differ-ential operator with polynomial coefficients on E × E × E and dependingpolynomially in β , and b is a polynomial in three complex variables. Suchidentities are known to exist (see [19], [20]), but are in general very difficultto find. It turns out that, in the case at hand, it is possible to find suchidentities. The proof uses in a crucial way the covariance property of thekernel l β with respect to the conformal action of G on E .During the proof of some results, we will need to use the EuclideanFourier transform. So it requires to extend the trilinear form to the Schwartzspace S ( E × E × E ). This is merely routine. For the definition of invariance,one should formulate the condition in terms of the infinitesimal action of theconformal group. It is a classical computation (see e.g. [17]) and the actionof the Lie algebra g = so (1 , d + 1) or of the universal enveloping algebra isby differential operators with polynomial coefficients. Hence they operateon the Schwartz space S ( E ). Moreover the meromorphic continuation of ℓ β yields tempered distributions (cf [10] for similar examples). Details are leftto the reader. Lemma 4.1.
Let M be the operator on S ( E × E ) given by M ϕ ( y, z ) = | y − z | ϕ ( y, z ) . Let λ, µ be two complex parameters.Then M is an intertwining operator for ( π λ ⊗ π µ , π λ − ⊗ π µ − ) .Proof. Let g ∈ G , and ϕ ∈ S ( E × E ), and assume that g is defined on aneighbourhood of Supp ( ϕ ). Then[ M ◦ (cid:0) π λ ( g ) ⊗ π µ ( g ) (cid:1) ] ϕ ( y, z ) = | y − z | κ ( g − , y ) ρ + λ κ ( g − , z ) ρ + µ ϕ ( g − ( y ) , g − ( z ))18 | g − ( y ) − g − ( z ) | κ ( g − , y ) ρ + λ − κ ( g − , z ) ρ + µ − ϕ ( g − ( y ) , g − ( z ))= [ π λ − ( g ) ⊗ π µ − ( g )]( M ϕ ) ( y, z ) . Introduce now the Knapp-Stein intertwining operator. For ν a complexparameter, let I ν be the operator on S ( E ) given by I ν ( f )( x ) = Z E | x − y | − d + ν f ( y ) dy . For ℜ ν >
0, the integral is convergent and defines a continuous operatoron S ( E ). It can be meromorphically continued to C , with simple poles at ν = − k, k ∈ N . It satisfies the following intertwining property I ν ◦ π ν ( g ) = π − ν ( g ) ◦ I ν . (27)Now, for λ, µ two complex parameters, form the operator N λ,µ = I − λ − ⊗ I − µ − ◦ M ◦ I µ ⊗ I µ S ( E × E ) I ν ⊗ I µ −−−−−→ S ( E × E ) M −→ S ( E × E ) I − λ − ⊗ I − µ − −−−−−−−−−−→ S ( E × E ) . For generic values of the parameters ( λ, µ ), N λ,µ is a well defined operatoron S ( E × E ), which, by construction intertwines the representation π λ ⊗ π µ and π λ +1 ⊗ π µ +1 .Let λ = ( λ , λ , λ ) be a generic triple in C , let β = ( β , β , β ) be thetriplet associated to λ through (23). Observe that β + 2 is associated tothe triple ( λ , λ + 1 , λ + 1).Consider the continuous trilinear form L on C ∞ c ( E ) × C ∞ c ( E ) × C ∞ c ( E )given by L ( f , f , f ) = L β +2 ( f ⊗ N λ ,λ ( f ⊗ f )) . From the intertwining property of N λ ,λ L ( π λ ( g ) f , π λ ( g ) f , π λ ( g ) f ) = L β +2 ( π λ ( g ) f ⊗ N λ ,λ [ π λ ( g ) f ⊗ π λ ( g ) f ])= L β +2 ( π λ ( g ) f ⊗ [ π λ +1 ( g ) ⊗ π λ +1 ( g )] ◦ N λ ,λ [ f ⊗ f ])= L β +2 ( f ⊗ N λ ,λ ( f ⊗ f ))= L ( f , f , f ) ,
19o that the form L is invariant w.r.t. ( π λ , π λ , π λ ). By the generic unique-ness result on the invariant trilinear form (see [3]), the form L has to beproportional to L β . Hence there exists a constant e = e ( β ) such that( N λ ,λ ) t (cid:0) l β +2 (cid:1) = e ( β ) l β . (28)As we will see now, the operator N λ,µ (hence also its transpose) is a differ-ential operator on E × E , so that (28) is indeed a Bernstein-Sato identity. Proposition 4.1.
For λ, µ ∈ C , let E λ,µ be the differential operator on E × E defined by E λ,µ = | y − z | ∆ y ∆ z − µ d X j =1 ( z j − y j ) ∂∂z j ∆ y − λ d X j =1 ( y j − z j ) ∂∂y j ∆ z + 2 µ (2 µ + 2 − d )∆ y + 2 λ (2 λ + 2 − d ))∆ z − λµ d X j =1 ∂∂y j ∂∂z j . (29) Its transpose F λ,µ = E tλ,µ is given by F λ,µ = | y − z | ∆ y ∆ z + 4( µ + 1) d X j =1 ( z j − y j ) ∂∂z j ∆ y + 4( λ + 1) d X j =1 ( y j − z j ) ∂∂y j ∆ z + 4( µ + 1)( µ + ρ )∆ y + 4( λ + 1)( λ + ρ )∆ z − λ + 1)( µ + 1) d X j =1 ∂∂y j ∂∂z j (30) The operator N λ,µ (for generic ( λ, µ ) ) is a differential operator on E × E ,and is given by N λ,µ = c ( λ, µ ) F λ,µ , where c ( λ, µ ) = π d
16 Γ( λ )Γ( − λ − µ )Γ( − µ − ρ − λ )Γ( ρ + λ + 1)Γ( ρ − µ )Γ( ρ + µ + 1) . roof. Introduce the Fourier transform on E , defined by F f ( ξ ) = ˆ f ( ξ ) = Z E × E e − i ( ξ,x ) f ( x ) dx , and extend it by duality to S ′ ( E ). The Fourier transform on E × E is definedaccordingly. Observe that I ν is a convolution operator with a tempereddistribution, so that the Fourier transform of I ν f is given by the product ofthe Fourier transform, i.e. F ( I ν f )( ξ ) = c ( ν ) | ξ | − ν ˆ f ( ξ )where c ( ν ) = 2 ν π d Γ( ν )Γ( d − ν )(see e.g. [10]).Next, as M acts by multiplication by a polynomial, the Fourier transformof M ϕ is given by F ( M ϕ )( ξ, η ) = ( − ∆ ξ + 2 R − ∆ η ) ˆ ϕ ( ξ, η ) , where ∆ is the Laplacian on E and R is the differential operator on S ( E × E )defined by Rϕ ( ξ, η ) = d X j =1 ∂ ϕ∂ξ j ∂η j . To prove the formula, it is enough to prove it for functions ϕ = f ⊗ g , where f, g ∈ S ( E ). Now ∂∂ξ j ( | ξ | − λ ˆ f ( ξ )) = | ξ | − λ ∂ ˆ f∂ξ j − λ | ξ | − λ − ξ j ˆ f ( ξ ) , so that ∆ ξ ( | ξ | − λ ˆ f ( ξ )) = | ξ | − λ ∆ ˆ f ( ξ ) − λ | ξ | − λ − d X j =1 ξ j ∂ ˆ f∂ξ j + 2 λ (2 λ + 2 − d ) | ξ | − λ − ˆ f ( ξ )21nd R ( | ξ | − λ ˆ f ( ξ ) | η | − µ ˆ g ( η )) = d X j =1 | ξ | − λ ∂ ˆ f∂ξ j | η | − µ ∂ ˆ g∂η j − λ d X j =1 ξ j | ξ | − λ − ˆ f ( ξ ) | η | − µ ∂ ˆ g∂η j − µ d X j =1 | ξ | − λ ∂ ˆ f∂ξ j η j | η | − µ − ˆ g ( η )+ 4 λµ | ξ | − λ − | η | − µ − ( d X j =1 ξ j η j ) ˆ f ( ξ )ˆ g ( η ) . Let apart the factor c (2 λ ) c (2 µ ) c ( − λ − c ( − µ − N λ,µ ( f ⊗ g ) is given by − | ξ | ∆ ξ ˆ f ( ξ ) | η | ˆ g ( η ) + 4 λ d X j =1 ξ j ∂ ˆ f∂ξ j ( ξ ) | η | ˆ g ( η ) − λ (2 λ + 2 − d ) ˆ f ( ξ ) | η | ˆ g ( η ) + 2 X j | ξ | ∂ ˆ f∂ξ j ( ξ ) | η | ∂ ˆ g∂η j ( η ) − λ d X j =1 ξ j ˆ f ( ξ ) | η | ∂ ˆ g∂η j ( η ) − µ X j =1 | ξ | ∂ ˆ f∂ξ j ( ξ ) η j ˆ g ( η )+ 8 λµ d X j =1 ξ j ˆ f ( ξ ) η j ˆ g ( η ) − | ξ | ˆ f ( ξ ) | η | ∆ η ˆ g ( η )+ 4 µ d X j =1 | ξ | ˆ f ( ξ ) η j ∂ ˆ g∂η j ( η ) − µ (2 µ + 2 − d ) | ξ | ˆ f ( ξ )ˆ g ( η ) . Now use the classical formulæ d ∂f∂y j ( ξ ) = iξ j ˆ f ( ξ ) [ ( y j f )( ξ ) = i ∂ ˆ f∂ξ j ( ξ ) c ∆ f ( ξ ) = −| ξ | ˆ f ( ξ ) \ | y | f ( y )( ξ ) = − ∆ ˆ f ( ξ )to obtain the following expression for N λ,µ ( f ⊗ g ) (up to the factor c (2 λ ) c (2 µ ) c ( − λ − c ( − µ − y ( | y | f )∆ z g + 4 λ d X j =1 ∂∂y j ( y j f )∆ z g + 2 λ (2 λ + 2 − d ) f ∆ z g − d X j =1 ∆ y ( y j f )∆ z ( z j g ) − λ d X j =1 ∂∂y j f ∆ z ( z j g ) − µ d X j =1 ∆ y ( y j f ) ∂∂z j g − λµ d X j =1 ∂∂y j f ∂∂z j g + ∆ y f ∆ z ( | z | g )+ 4 µ d X j =1 ∆ y f ∂∂z j ( z j g ) + 2 µ (2 µ + 2 − d )∆ y f g The final expression for N λ,µ and ( N λ,µ ) t follows easily.As announced, E λ ,λ (being proportional to N tλ ,λ )is a candidate fora Bernstein-Sato identity for the kernel ℓ β (the λ ’s being related to β by(23)). By brute force computation, the following identity is obtained. Theorem 4.1 (Bernstein-Sato identity) . For β = ( β , β , β ) in C , let B β be the following differential operator on E × EB β = | y − z | ∆ y ∆ z + 2( β + β + d ) d X j =1 ( z j − y j ) ∂∂y j ∆ z + 2( β + β + d ) d X j =1 ( y j − z j ) ∂∂z j ∆ y + ( β + β + d )( β + β + 2)∆ z + ( β + β + d )( β + β + 2)∆ y − β + β + d )( β + β + d ) d X j =1 ∂ ∂y j ∂z j . Then B β l β +2 = b ( β ) l β (31) where b ( β ) = ( β + d )( β + 2)( β + β + β + 2 d )( β + β + β + d + 2) . Applications of the Bernstein-Sato identity
The first application of the Bernstein-Sato identity is the computation ofthe residues of the distribution L β along the plane H k by induction over k . Proposition 5.1.
Let β be such that β + β + β = − d − k − for some k ∈ N . Assume that β j / ∈ − d − N ( j = 1 , , ), and β = − . Then, for f ∈ S ( E ) and g ∈ S ( E × E ) , Res ( L β , β )( f ⊗ g (cid:1) = 1(2 k + 2)(2 k + d )( β + 2)( β + d ) Res ( L β , β +2 )( f ⊗ B t β g (cid:1) (32) Proof.
For generic values of β , from the Bernstein-Sato identitiy, L β ( f ⊗ g ) = ( ℓ β , f ⊗ g ) = 1 b ( β ) ( B β ℓ β +2 , f ⊗ g ) = 1 b ( β ) ( ℓ β +2 , f ⊗ B t β g ) , and compute the residue at β on both sides.Let C β = B t β . Except for the change of parameters, this is nothing butthe operator F λ,µ . Proposition 5.2. C β = B t β = | y − z | ∆ y ∆ z + 2( β + β + d + 2) d X j =1 ( z j − y j ) ∂∂z j ∆ y + 2( β + β + d + 2) d X j =1 ( y j − z j ) ∂∂y j ∆ z + ( β + β + 2 d )( β + β + d + 2)∆ y − β + β + d + 2)( β + β + d + 2) d X j =1 ∂ ∂y j ∂z j + ( β + β + 2 d )( β + β + d + 2)∆ z . (33)To write an expression for the residue at a pole in H k , where k ∈ N , letuse the following convention: for β = ( β , β , β ) and k ∈ N , let β − (2 k ) = ( β − k, β , β )Now, for β ∈ H , define the differential operator E ( k ) β on E × E by C (0) β = Id , C ( k ) β = C β − ◦ · · · ◦ C β − (2 k ) heorem 5.1. Let β ∈ H , and let k ∈ N . Assume that β j / ∈ − d − N ( j = 1 , , and β / ∈ { , , . . . , k − } . Then Res ( L β , β − (2 k ) )( f ⊗ g ) = c k ( β ) Z E f ( x ) (cid:0) C ( k ) β g (cid:1) ( x, x ) dx (34) where c k ( β ) = 116 k k ! 1( ρ ) k − β ) k − β − ρ + 1) k c ( β ) . The Bernstein-Sato operator can also be used to describe a family ofcovariant bidifferential operators. Let λ, µ be in C . For k ∈ N , let F ( k ) λ,µ bethe bidifferential operator defined by F ( k ) λ,µ f ( x ) = F λ + k − , µ + k − ◦ · · · ◦ F λ,µ f ( x, x ) , (35)where F λ,µ is the differential operator on E × E defined by (30). Theorem 5.2.
Let λ, µ in C , and k ∈ N . Then the operator F ( k ) λ,µ is confor-mally covariant with respect to ( π λ ⊗ π µ , π λ + µ + ρ +2 k ) .Proof. Recall that the operator F λ,µ is covariant w.r.t. ( π λ ⊗ π µ , π λ +1 ⊗ π µ +1 ).So, by induction, F λ + k − , µ + k − ◦· · · ◦ F λµ is covariant w.r.t. ( π λ ⊗ π µ , π λ + k ⊗ π µ + k ). Now the map C ∞ c ( E × E ) ∋ f f ∈ C ∞ c ( E ) , where f ( x ) = f ( x, x ), is covariant w.r.t. ( π λ + k ⊗ π µ + k , π λ + µ + ρ +2 k ). Theassertion follows.For k = 1, one gets F (1) λ, µ = 4( µ + 1)( µ + ρ )∆ y − λ + 1)( µ + 1) R + 4( λ + 1)( λ + ρ )∆ z to be compared with (26). The construction of the covariant differential operator N λ,µ admits anatural generalization. Let τ be the standard representation of G and τ ′ itsdual representation. Choose highest weight vectors v and φ (with respect to25ome suitable ordering) for τ and τ ′ , respectively. Then up to some constant,the multiplication by the matrix coefficient g , g (cid:10) τ ( g ) v, τ ′ ( g ) φ (cid:11) coincides with M from lemma 4.1. Upon replacing τ by another irreduciblefinite-dimensional representation of G one obtains a multiplication operatorthat intertwines the tensor products of general (i.e. not necessarily spherical)principle series representations (see [1]). Using appropriate intertwiningoperators one obtains an associated differential operator. We would also liketo point out that the operators used by Oksak in his work [16] on invarianttrilinear forms for G = Sl ( C ) ≃ Spin (3 ,
1) are of this type. See also [12]. A byproduct of Proposition 2.1 is that the residue at a point β ∈ H vanishes identically if − β − d ∈ − N , i.e. if β ∈ − d + 2 m, m ∈ N . For such avalue, observe that λ = β + β ρ = − β − d ρ = − m . Now for m ≥
1, there exists a nontrivial differential operator R m (see [2])on C ∞ c ( E ) which is covariant w.r.t. ( π − m , π m ). Now let˜ β = ( β − m, β + 2 m, β + 2 m ) , so that e λ = ( m, λ , λ ) . As e β + e β + e β = − d + 2 m , e β is no longer a pole. So the form L e β is welldefined, and the form ( f , f , f ) e β ( R m f , f , f )is invariant with respect to ( π − m , π λ , π λ ). However, the relation λ + λ + λ = − ρ guarantees that the form( f , f , f ) Z E f ( x ) f ( x ) f ( x ) dx is invariant under ( π λ , π λ , π λ ). So, for λ = − m and ( λ , λ ) generic,we have produced two (linearly independant) trilinear invariant forms on C ∞ c ( E ) ×C ∞ c ( E ) ×C ∞ c ( E ) w.r.t. ( π − m , π λ , π λ ) . Although we won’t developthese aspects here, the same remark can be used to produce, for specific val-ues of λ , two (linearly independant) bidifferential operators covariant under26he same actions of the conformal group. Notice that this is in concordancewith the results and the philosophy of [13] and [14]. The relation between our formula for covariant bidifferential operators(35) and the formulæ obtained in [17] or in [14] is still to be investigated,and the coefficients which relate them seem to be important. In the classicalsetting (i.e. for the original Rankin-Cohen operators acting on the upperhalf-plane), much effort has been devoted to understand the structure of thisfamily of operators (see [23], [5], [22], [6], [18]). We hope that our realizationof these operators will add to the understanding of the family of generalizedRankin Cohen operators.
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Addresses(RB) Mathematisches Institut Universit¨at T¨ubingen, Auf der Morgenstelle 10, 72076T¨ubingen, Germany(JLC) Institut ´Elie Cartan, Universit´e Henri Poincar´e (Nancy 1), 54506 Vandoeuvre-l`es-Nancy, France.